# Upper join-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties

## Contents

## Definition

### Definition with symbols

A subgroup property is said to be **upper join-closed** if given and are intermediate subgroups of containing (indexed by a nonempty set ) and satisfies in each , we have that satisfies in the join of subgroups .

## Relation with other metaproperties

### Stronger metaproperties

- Lower-intersection upper-join closed subgroup property
- LU-join closed subgroup property
- Upward-closed subgroup property
- Izable subgroup property

### Weaker metaproperties

## Related notions

Given a subgroup property that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup of associate a unique largest subgroup containing for which satisfies in .

Such a subgroup property is termed an izable subgroup property and the that we get is termed the izing subgroup of for that subgroup property.

## Properties satisfying it

### Normality

Normality is an upper join-closed subgroup property, viz, if and are intermediate subgroups such that and , then .

### Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.